isogonal trajectory

Let a one-parametric family of plane curves$\gamma$  have the differential equation

 $\displaystyle F(x,\,y,\,\frac{dy}{dx})\;=\;0.$ (1)

We want to determine the isogonal trajectories of this family, i.e. the curves  $\iota$  intersecting all members of the family under a given angle, which is denoted by $\omega$. For this purpose, we denote the slope angle of any curve $\gamma$ at such an intersection point by $\alpha$ and the slope angle of $\iota$ at the same point by $\beta$.  Then

 $\beta-\alpha\;=\;\omega\quad(\mbox{or alternatively\;\;}-\omega),$

and accordingly

 $\frac{dy}{dx}\;=\;\tan\alpha\;=\;\frac{\tan\beta-\tan\omega}{1+\tan\beta\tan% \omega}\;=\;\frac{y^{\prime}-\tan\omega}{1+y^{\prime}\tan\omega},$

where $y^{\prime}$ means the slope of $\iota$.  Thus the equation

 $\displaystyle F(x,\,y,\,\frac{y^{\prime}-\tan\omega}{1+y^{\prime}\tan\omega})% \;=\;0$ (2)

is satisfied by the derivative $y^{\prime}$ of the ordinate of $\iota$.  In other , (2) is the differential equation of all isogonal trajectories of the given family of curves.

Note.  In the special case  $\omega=\frac{\pi}{2}$,  it’s a question of orthogonal trajectories.

 Title isogonal trajectory Canonical name IsogonalTrajectory Date of creation 2013-03-22 18:59:20 Last modified on 2013-03-22 18:59:20 Owner pahio (2872) Last modified by pahio (2872) Numerical id 7 Author pahio (2872) Entry type Derivation Classification msc 51N20 Classification msc 34A26 Classification msc 34A09 Related topic AngleBetweenTwoCurves Related topic OrthogonalCurves Related topic ExampleOfIsogonalTrajectory Related topic AngleBetweenTwoLines Defines isogonal trajectory